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How to Tell One Differential Equation from Another

Part of the Differential Equations Workbook For Dummies Cheat Sheet

Before you can solve a differential equation, you need to know what kind it is. There are several different types of equations, including linear, separable, exact, homogeneous, and nonhomogeneous.

Linear differential equations deal solely with derivatives to the first power (forget about derivatives raised to any higher power).

The power referred to here is the power the derivative is raised to, not the order of the derivative. Here’s a pretty typical-looking linear differential equation:

image0.png

Separable differential equations can be written so that all terms in x and all terms in y appear on opposite sides of the equation, as you can see in this example:

image1.png

which can also be written as

image2.png

Exact differential equations are those where you can find a function whose partial derivatives correspond to the terms in the differential equation. Here’s an example:

image3.png

Homogeneous differential equations contain only derivatives of y and terms involving y. As you can see in this equation, they’re also set to 0:

image4.png

Nonhomogeneous differential equations are the same as homogeneous differential equations but with one exception: They can only have terms involving x and/or constants on the right side. Here’s an example of a nonhomogeneous differential equation:

image5.png

The general solution of this nonhomogeneous differential equation:

image6.png

is

image7.png

where c1y1(x) + c2y2(x) is the general solution of the corresponding homogeneous differential equation

image8.png

and yp(x) is a particular solution to the nonhomogeneous equation.

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